arXiv subjects
Publications and source records attributed to Benny Sudakov.
In this short note we prove that there is a constant $c$ such that every k-edge-coloring of the complete graph K_n with n > 2^{ck} contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and is the first exponential bound for this problem.
For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.
Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value of |C(G)| over all graphs G of average degree d and girth g. Erdos conjectured that |C(G)| =Ω(d^{\lfloor (g-1)/2\rfloor}) for all such graphs, and we prove this conjecture. In particular, the longest cycle in a graph of average degree d and girth g has length Ω(d^{\lfloor (g-1)/2\rfloor}). The study of this problem was initiated by Ore in 1967 and our result improves all previously known lower bounds on the length of the longest cycle. Moreover, our bound cannot be improved in general, since known constructions of d-regular Moore Graphs of girth g have roughly that many vertices. We also show that Ω(d^{\lfloor (g-1)/2\rfloor}) is a lower bound for the number of odd cycle lengths in a graph of chromatic number d and girth g. Further results are obtained for the number of cycle lengths in H-free graphs of average degree d. In the second part of the paper, motivated by the conjecture of Erdos and Gyarfas that every graph of minimum degree at least three contains a cycle of length a power of two, we prove a general theorem which gives an upper bound on the average degree of an n-vertex graph with no cycle of even length in a prescribed infinite sequence of integers. For many sequences, including the powers of two, our theorem gives the upper bound e^{O(\log^* n)} on the average degree of graph of order n with no cycle of length in the sequence, where \log^* n is the number of times the binary logarithm must be applied to n to get a number which is at most
In this paper we show how to find nearly optimal embeddings of large trees in several natural classes of graphs. The size of the tree T can be as large as a constant fraction of the size of the graph G, and the maximum degree of T can be close to the minimum degree of G. For example, we prove that any graph of minimum degree d without 4-cycles contains every tree of size εd^2 and maximum degree at most (1-2ε)d - 2. As there exist d-regular graphs without 4-cycles of size O(d^2), this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth, graphs with no complete bipartite subgraph K_{s,t}, random and certain pseudorandom graphs. These results are obtained using a simple and very natural randomized embedding algorithm, which can be viewed as a "self-avoiding tree-indexed random walk".
Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number χ_g(G) is the minimum k for which the first player has a winning strategy. In this paper we analyze the asymptotic behavior of this parameter for a random graph G_{n,p}. We show that with high probability the game chromatic number of G_{n,p} is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs.
We study a model of random uniform hypergraphs, where a random instance is obtained by adding random edges to a large hypergraph of a given density. We obtain a tight bound on the number of random edges required to ensure non-2-colorability. We prove that for any k-uniform hypergraph with Omega(n^{k-epsilon}) edges, adding omega(n^{k epsilon/2}) random edges makes the hypergraph almost surely non-2-colorable. This is essentially tight, since there is a 2-colorable hypergraph with Omega(n^{k-ε}) edges which almost surely remains 2-colorable even after adding o(n^{k ε/ 2}) random edges.
Extending several previous results we obtained nearly tight estimates on the maximum size of a clique-minor in various classes of expanding graphs. These results can be used to show that graphs without short cycles and other H-free graphs contain large clique-minors, resolving some open questions in this area.
A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E_P(G). Our first result states that for any monotone graph property P, any ε>0 and n-vertex input graph G one can approximate E_P(G) up to an additive error of εn^2 Our second main result shows that such approximation is essentially best possible and for most properties, it is NP-hard to approximate E_P(G) up to an additive error of n^{2-δ}, for any fixed positive δ. The proof requires several new combinatorial ideas and involves tools from Extremal Graph Theory together with spectral techniques. Interestingly, prior to this work it was not even known that computing E_P(G) precisely for dense monotone properties is NP-hard. We thus answer (in a strong form) a question of Yannakakis raised in 1981.
Let G be a graph on n vertices in which every induced subgraph on s=\log^3 n vertices has an independent set of size at least t=\log n. What is the largest q=q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdos and Hajnal. We show that q(n)=Θ(\log^2 n/\log \log n), investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey-type problem.
We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-ε)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2 and 0<ε<1, there exists a constant c=c(d,ε) such that a random graph G(n,c/n) contains almost surely a copy of every tree T on (1-ε)n vertices with maximum degree at most d. We also prove that if an (n,D,λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λin their absolute values) has large enough spectral gap D/λas a function of d and ε, then G has a copy of every tree T as above.
We show that every K_4-free graph G with n vertices can be made bipartite by deleting at most n^2/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdos.
For two graph H and G, the Ramsey number r(H, G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph K_n on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions of Erdos and Harary, in this note we study how the Ramsey number r(K_s, G) depends on the size of the graph G. For s \geq 3, we prove that for every G with m edges, r(K_s,G) \geq c (m/\log m)^{\frac{s+1}{s+3}} for some positive constant c depending only on s. This lower bound improves an earlier result of Erdos, Faudree, Rousseau, and Schelp, and is tight up to a polylogarithmic factor when s=3. We also study the maximum value of r(K_s,G) as a function of m.
Let G be a graph with n vertices, and let k be an integer dividing n. G is said to be strongly k-colorable if for every partition of V(G) into disjoint sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex k-coloring of G with each color appearing exactly once in each V_i. In the case when k does not divide n, G is defined to be strongly k-colorable if the graph obtained by adding k \lceil n/k \rceil - n isolated vertices is strongly k-colorable. The strong chromatic number of G is the minimum k for which G is strongly k-colorable. In this paper, we study the behavior of this parameter for the random graph G(n, p). In the dense case when p >> n^{-1/3}, we prove that the strong chromatic number is a.s. concentrated on one value Δ+1, where Δis the maximum degree of the graph. We also obtain several weaker results for sparse random graphs.
Let G be a graph with maximum degree Δwhose vertex set is partitioned into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G) containing exactly one vertex from each part V_i. If it is also an independent set, then we call it an independent transversal. The local degree of G is the maximum number of neighbors of a vertex v in a part V_i, taken over all choices of V_i and v \not \in V_i. We prove that for every fixed ε> 0, if all part sizes |V_i| >= (1+ε)Δand the local degree of G is o(Δ), then G has an independent transversal for sufficiently large Δ. This extends several previous results and settles (in a stronger form) a conjecture of Aharoni and Holzman. We then generalize this result to transversals that induce no cliques of size s. (Note that independent transversals correspond to s=2.) In that context, we prove that parts of size |V_i| >= (1+ε)[Δ/(s-1)] and local degree o(Δ) guarantee the existence of such a transversal, and we provide a construction that shows this is asymptotically tight.
Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate. Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs and the aim of this survey is to provide a systematic treatment of this concept.
Ohba has conjectured \cite{ohb} that if the graph $G$ has $2χ(G)+1$ or fewer vertices then the list chromatic number and chromatic number of $G$ are equal. In this paper we prove that this conjecture is asymptotically correct. More precisely we obtain that for any $0<ε<1$, there exist an $n_0=n_0(ε)$ such that the list chromatic number of $G$ equals its chromatic number, provided $$n_0 \leq |V(G) | \le (2-ε)χ(G).$$
Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p. We prove that the largest eigenvalue of the adjacency matrix of G is almost surely λ_1(G)= (1+o(1)) max(Δ^{1/2}(G),np), where Δ(G) is the maximum degree of G and o(1) term tends to zero as max (Δ^{1/2}(G), np) tends to infinity.
Let $C^{2k}_r$ be the $2k$-uniform hypergraph obtained by letting $P_1,...,P_r$ be pairwise disjoint sets of size $k$ and taking as edges all sets $P_i \cup P_j$ with $i \neq j$. This can be thought of as the `$k$-expansion' of the complete graph $K_r$: each vertex has been replaced with a set of size $k$. We determine the exact Turan number of $C^{2k}_3$ and the corresponding extremal hypergraph, thus confirming a conjecture of Frankl. Sidorenko has given an upper bound of $(r-2) / (r-1)$ for the Turán density of $C^{2k}_r$ for any $r$, and a construction establishing a matching lower bound when $r$ is of the form $2^p + 1$. We show that when $r = 2^p + 1$, any $C^4_r$-free hypergraph of density $(r-2)/(r-1) - o(1)$ looks approximately like Sidorenko's construction. On the other hand, when $r$ is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of $C^4_r$ to $(r-2)/(r-1) - c(r)$, where $c(r)$ is a constant depending only on $r$. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal-Katona theorem and properties of Krawtchouck polynomials.